The Poisson window (or more generically exponential window) can be written
The Poisson window is plotted in Fig.3.19. In the plane, the Poisson window has the effect of radially contracting the unit circle. Consider an infinitely long Poisson window (no truncation by a rectangular window ) applied to a causal signal having transform :
Thus, the unit-circle response is moved to . This means, for example, that marginally stable poles in now decay as in .
The effect of this radial -plane contraction is shown in Fig.3.20.
The Poisson window can be useful for impulse-response modeling by poles and/or zeros (``system identification''). In such applications, the window length is best chosen to include substantially all of the impulse-response data.